Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experimental tools to probe these phenomena.
In particle physics, supersymmetry (SUSY) is a principle that proposes a relationship between two basic classes of elementary particles: bosons, which have an integer-valued spin, and fermions, which have a half-integer spin. A type of spacetime symmetry, supersymmetry is a possible candidate for undiscovered particle physics, and seen as an elegant solution to many current problems in particle physics if confirmed correct, which could resolve various areas where current theories are believed to be incomplete. A supersymmetrical extension to the Standard Model would resolve major hierarchy problems within gauge theory, by guaranteeing that quadratic divergences of all orders will cancel out in perturbation theory.
In geometry and physics, spinors are elements of a (complex) vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight (infinitesimal) rotation. However, when a sequence of such small rotations is composed (integrated) to form an overall final rotation, the resulting spinor transformation depends on which sequence of small rotations was used: unlike vectors and tensors, a spinor transforms to its negative when the space is rotated through a complete turn from 0° to 360°. This property characterizes spinors. It is also possible to associate a substantially similar notion of spinor to Minkowski space in which case the Lorentz transformations of special relativity play the role of rotations. Spinors were introduced in geometry by Élie Cartan in 1913. In the 1920s physicists discovered that spinors are essential to describe the intrinsic angular momentum, or "spin", of the electron and other subatomic particles.
Extended supersymmetry is also called , supersymmetry, for example. Extended supersymmetry is very important for analysis of mathematical properties of quantum field theory and superstring theory. The more extended supersymmetry is, the more it constrains physical observables and parameters.
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics and is used to construct physical models of subatomic particles and quasiparticles.
Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modeling them as vibrations of tiny supersymmetric strings.
In theoretical physics, a supersymmetry algebra is a mathematical formalism for describing the relation between bosons and fermions. The supersymmetry algebra contains not only the Poincaré algebra and a compact subalgebra of internal symmetries, but also contains some fermionic supercharges, transforming as a sum of N real spinor representations of the Poincaré group. Such symmetries are allowed by the Haag–Łopuszański–Sohnius theorem. When N>1 the algebra is said to have extended supersymmetry. The supersymmetry algebra is a semidirect sum of a central extension of the super-Poincaré algebra by a compact Lie algebra B of internal symmetries.
In supersymmetry, harmonic superspace is one way of dealing with supersymmetric theories with 8 real SUSY generators in a manifestly covariant manner. It turns out that the 8 real SUSY generators are pseudoreal, and after complexification, correspond to the tensor product of a four-dimensional Dirac spinor with the fundamental representation of SU(2)R. The quotient space , which is a 2-sphere/Riemann sphere.
In supersymmetry, a theory of particle physics, projective superspace is one way of dealing with supersymmetric theories, i.e. with 8 real SUSY generators, in a manifestly covariant manner.
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Noether's (first) theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether in 1915 and published in 1918, after a special case was proven by E. Cosserat & F. Cosserat in 1909. The action of a physical system is the integral over time of a Lagrangian function, from which the system's behavior can be determined by the principle of least action. This theorem only applies to continuous and smooth symmetries over physical space.
In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2-grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. In most of these theories, the even elements of the superalgebra correspond to bosons and odd elements to fermions.
In mathematics and physics, n-dimensional anti-de Sitter space (AdSn) is a maximally symmetric Lorentzian manifold with constant negative scalar curvature. Anti-de Sitter space and de Sitter space are named after Willem de Sitter (1872–1934), professor of astronomy at Leiden University and director of the Leiden Observatory. Willem de Sitter and Albert Einstein worked together closely in Leiden in the 1920s on the spacetime structure of the universe.
Superspace is the coordinate space of a theory exhibiting supersymmetry. In such a formulation, along with ordinary space dimensions x, y, z, ..., there are also "anticommuting" dimensions whose coordinates are labeled in Grassmann numbers rather than real numbers. The ordinary space dimensions correspond to bosonic degrees of freedom, the anticommuting dimensions to fermionic degrees of freedom.
In theoretical physics, supergravity is a modern field theory that combines the principles of supersymmetry and general relativity; in contrast to non-gravitational supersymmetric theories such as the Minimal Supersymmetric Standard Model. Supergravity is the gauge theory of local supersymmetry. Since the supersymmetry (SUSY) generators form together with the Poincaré algebra a superalgebra, called the super-Poincaré algebra, gauging supersymmetry makes gravity arise in a natural way.
The Minimal Supersymmetric Standard Model (MSSM) is an extension to the Standard Model that realizes supersymmetry. MSSM is the minimal supersymmetrical model as it considers only "the [minimum] number of new particle states and new interactions consistent with phenomenology". Supersymmetry pairs bosons with fermions, so every Standard Model particle has a superpartner yet undiscovered. If we find these superparticles, it equates to discovering such particles as dark matter, could provide evidence for grand unification, and provide hints as to whether string theory describes nature. The failure to find evidence for supersymmetry using the Large Hadron Collider suggests a leaning to abandon it.
In theoretical physics, a supermultiplet is a representation of a supersymmetry algebra. It consists of a collection of particles, called superpartners, corresponding to operators in a quantum field theory which in superspace are represented by superfields.
In theoretical physics, a super-Poincaré algebra is an extension of the Poincaré algebra to incorporate supersymmetry, a relation between bosons and fermions. They are examples of supersymmetry algebras, and are Lie superalgebras. Thus a super-Poincaré algebra is a Z2-graded vector space with a graded Lie bracket such that the even part is a Lie algebra containing the Poincaré algebra, and the odd part is built from spinors on which there is an anticommutation relation with values in the even part.
In general relativity, the Gibbons–Hawking–York boundary term is a term that needs to be added to the Einstein–Hilbert action when the underlying spacetime manifold has a boundary.
In theoretical physics, there are many theories with supersymmetry (SUSY) which also have internal gauge symmetries. Supersymmetric gauge theory generalizes this notion.
In theoretical physics, the superconformal algebra is a graded Lie algebra or superalgebra that combines the conformal algebra and supersymmetry. In two dimensions, the superconformal algebra is infinite-dimensional. In higher dimensions, superconformal algebras are finite-dimensional and generate the superconformal group.
In set theory and computability theory, Kleene's
is a canonical subset of the natural numbers when regarded as ordinal notations. It contains ordinal notations for every recursive ordinal, that is, ordinals below Church–Kleene ordinal,
. Since
is the first ordinal not representable in a computable system of ordinal notations the elements of
can be regarded as the canonical ordinal notations.
In theoretical physics a nonrenormalization theorem is a limitation on how a certain quantity in the classical description of a quantum field theory may be modified by renormalization in the full quantum theory. Renormalization theorems are common in theories with a sufficient amount of supersymmetry, usually at least 4 supercharges.
Learning with errors (LWE) is a problem in machine learning that is conjectured to be hard to solve. Introduced by Oded Regev in 2005, it is a generalization of the parity learning problem. Regev showed, furthermore, that the LWE problem is as hard to solve as several worst-case lattice problems. The LWE problem has recently been used as a hardness assumption to create public-key cryptosystems, such as the ring learning with errors key exchange by Peikert.
In physics, Liouville field theory is a two-dimensional conformal field theory whose classical equation of motion is a generalization of Liouville's equation.
In algebra, the Nichols algebra of a braided vector space is a braided Hopf algebra which is denoted by
and named after the mathematician Warren Nichols. It takes the role of quantum Borel part of a pointed Hopf algebra such as a quantum groups and their well known finite-dimensional truncations. Nichols algebras can immediately be used to write down new such quantum groups by using the Radford biproduct.
In theoretical particle physics, the gluon field strength tensor is a second order tensor field characterizing the gluon interaction between quarks.
Supersymmetric theory of stochastic dynamics or stochastics (STS) is an exact theory of stochastic (partial) differential equations (SDEs), the class of mathematical models with the widest applicability covering, in particular, all continuous time dynamical systems, with and without noise. The main utility of the theory from the physical point of view is a rigorous theoretical explanation of the ubiquitous spontaneous long-range dynamical behavior that manifests itself across disciplines via such phenomena as 1/f, flicker, and crackling noises and the power-law statistics, or Zipf's law, of instantonic processes like earthquakes and neuroavalanches. From the mathematical point of view, STS is interesting because it bridges the two major parts of mathematical physics – the dynamical systems theory and topological field theories. Besides these and related disciplines such as algebraic topology and supersymmetric field theories, STS is also connected with the traditional theory of stochastic differential equations and the theory of pseudo-Hermitian operators.
